Questions on the factorial function, $n!=n\times(n-1)\times\cdots\times1$. Consider using the tag (gamma-function) if dealing with noninteger arguments.

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1answer
18 views

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear?

How many straight lines can be drawn between five points (A, B, C, D, and E), no three of which are colinear? Attempt: Given 5 points, a line consist always of 2 points. Thus the total number of ...
0
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1answer
20 views

How many zip codes are as large as 6000-0000, are even numbers, and have a 7 as their third digit?

When they were first introduced, postal zip codes were five digit numbers, theoretically ranging from $00000$ to $99999$. (In reality, the lowest zip code was $00601$ for San Juan, Puerto Rico; the ...
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3answers
29 views

In how many ways can the word ELEEMOSYNARY be arranged.

In how many ways can be the letters of the word ELEEMOSYNARY be arranged so that the S is always immediately followed by a Y? Attempt: There are 3 Es, and 2 Ys, and and then all letters appear once ...
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1answer
12 views

How many different eight note melodies within a single octave can be written if black/white keys alternate.

An octave contains 12 distinct notes(on a piano, five black keys and seven white keys). How many different eight notes melodies within a single octave can be written if the black keys and white keys ...
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2answers
20 views

A coke hand in bridge from deck of cards.

A coke hand in bridge is one where none of the thirteen cards is an an ace or is higher than a 9. What is the probability of being dealt such a hand? Attempt: Suppose the thirteens cards are amoung ...
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1answer
33 views

Five cards selected out of 52 cards. Find probalbilty sum of the faces is 48 or more.

Five cards are dealt from a standard 52 card deck. What is the probability that the sum of the faces on the five cards is 48 or more? Attempt: Five cards can be selected out of 52 cards by 52_C_5 ...
1
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1answer
39 views

What are chances that not all S's will be adjacent given a phrase at random.

IF the letters in the phrase A ROLLING STONE GATHERS NO MOSS are arranged at random, what are the chances that not all the S's will be adjacent. Attempt: Given there are 6 letters that appear twice, ...
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1answer
27 views

A bridge hand (13 cards) is dealt from a standard 52 card deck. Given events A and B, find $P(A \cup B)$.

A bridge hand (13 cards) is dealt from a standard 52 card deck. Let A be the event that the hand contains four aces. Let B be the event that the hand contains four kings. Find $P(A \cup B)$. Attempt: ...
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3answers
28 views

Why is $1 \times 3 \times 5 \times \cdots \times (2k-3) = \frac{(2k-2)!}{2^{(k-1)}(k-1)!}$

In order to find out the Catalan numbers from their generating function you have to evaluate the product above. Here is what I thought: \begin{align*} 1 \times 3 \times 5 \times...\times (2k-3) ...
1
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1answer
52 views

How many ways different sets of values can be chosen for the $x_s$ , if $x_1 + x_2 + x_3 = 20$?

Your statistics teacher announces a twenty-page reading assignment on Monday that is to be finished by Thursday morning. You intend to read the first $x_1$ pages Monday, the next $x_2$ pages ...
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1answer
38 views

How many ways can a twelve member cheerleading be pair up.

Problem: How many ways can a twelve member cheerleading squad(6 men and 6 women) pair up to form 6 male-female teams? What might the number 6!6!2^6 represent? What might the number 6!6!2^6*2^12 ...
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2answers
62 views

Prove $1(1!)+\dots+n(n!) = (n+1)!-1$ using induction

So I'm trying to prove this statement (through induction): $$1(1!)+2(2!)+\dots +n(n!)=(n+1)!-1$$ But I'm confused with the inductive step here: $$(n+1)!-1+[(n+1)(n+1)!] = (n+2)!-1$$ What do I do ...
4
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2answers
61 views

How find the positive numbers $n$ such that $n!=\overline{1999a_{1}a_{2}\cdots a_{k}\cdots}$

Question: find all the postive integer $n$ such $$n!=\overline{1999a_{1}a_{2}\cdots a_{k}\cdots}$$ where $a_{i}\in[0,9]$ (or mean $n!$ left-most four digits are $1999$) I think ...
5
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0answers
62 views

Double factorial as a sum

I believe the following equality to hold for all integer $l\geq 1$ $$(2l+1)!2^l\sum_{k=0}^l\frac{(-1)^k(l-k)!}{k!(2l-2k+1)!4^k}=(-1)^l(2l-1)!!$$ (it's correct for at least $l=1,2,3,4$), but cannot ...
0
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1answer
26 views

Solving inequalities involving factorials

I've been trying to prove a statement that I found in a book. Given that(I've already proven these statements): $$ \forall\; n\in Z^+:(1 + \frac 1n)^n = 1 + \sum_{k=1}^n \frac ...
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1answer
31 views

Show $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$

I am trying to show that $\frac{n}{2} \log(n!) = \Omega (n^2 \log(n))$ but I seem to get a conflicting result. What i did is: $n!=1*2*3*...*n \leq n*n*n*...*n=n^n$, so $\frac{n}{2} \log(n!) \leq ...
0
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1answer
30 views

Proving $\lg n!=\Omega(n\lg n)$

In the answer given in the book for the proof of $\lg n=\Omega(n\lg n)$ there are several steps which I don't understand . $$\lg n!=\lg n+\lg(n-1)+\lg(n-2)+ ....+\lg(2)+\lg 1$$ Then it says that ...
0
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1answer
28 views

Solving an equation involving factorial notation

I was given this problem in the text book: $$\frac{(n+4)!}{(n+2)!} = 6$$ $$n \in I $$ Since the textbook doesn't have the solution, I'm wondering if I'm right: $$\frac{(n+4)!}{(n+2)!} \Rightarrow ...
9
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6answers
400 views

The limit of $(n!)^{1/n}/n$ as $n\to\infty$ [duplicate]

(Proof necessary) $$\lim_{n \to \infty} \frac{(n!)^{\frac{1}{n}}}{n}$$ I don't have an answer yet, but I know it exists, and is less than $1$. Edit. Winther's answer is the most correct I don't ...
0
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2answers
35 views

Double factorial formula

I don't know why this is true: $$(2k+1)!!=\frac{(2k+1)!}{2^kk!}$$ Can anyone explain it for me? I know what is double factorial, but would like to know how this formula was derived. Thanks.
5
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2answers
74 views

Is $(\log(n))!$ a polynomially bounded function?

Is the following statement true? How would you prove it? i.e. Is it a polynomially bounded? $$ \lceil \lg(n) \rceil ! \in O(n^k) $$ How about $$ \lceil \lg \lg(n) \rceil ! \in O(n^k) $$ Thanks a ...
2
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1answer
80 views

What does the factorial of a negative number signify?

I understand that the factorial gives the number of arrangements. For example: the factorial of zero i.e. an empty set ( doesn't occur) is 1. As the empty set can be arranged only in 1 way - i.e. by ...
5
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2answers
71 views

Challenge: How to prove this identity between bi- and trinomial coefficients?

This question is the continuation of its predecessor. Using the convention that trinomial coefficients $$ \binom{n}{k_1,k_2,k_3}=\frac{n!}{k_1! k_2! k_3!} $$ are zero if $k_i<0$ or $\sum_i k_i\neq ...
2
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0answers
64 views

Checking whether it is integer or not.

I'm trying to prove following term is integer for all $m,n \in \mathbb{N} :$ $$\frac{2^{m+n-1}\prod\limits_{k=1}^m(4k-3)\prod\limits_{l=1}^n(4l-1)}{(m+n)!}$$ I checked that if $m=n$ then this term ...
0
votes
2answers
79 views

Prove that $6! \mid n(n+1)…(n+5)$ [closed]

Prove that for all $n \in \mathbb{Z}$, $6! \mid n\cdot(n+1)\cdots(n+5)$ using only criteria of divisibility (without using combinatorial arguments).
3
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0answers
69 views

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer

If $x$ and $y$ are positive integers then $\frac{(2x)!(2y)!}{x!y!(x+y)!}$ is an integer I have to show that the proposition above is true for any $x,y\in\mathbb{Z^+}$ by means of Legendre's ...
2
votes
2answers
41 views

Infinite sum of ratio of factorials

Mathematica tells me that for $r, p$ positive integers with $\mathcal{r} < p$ we have $$\sum_{i=p+1}^\infty \frac{(i-r)!}{(i+r)!} = \frac{(p+1-r)!}{(2r-1)(p+r)!}. $$ Can anyone point me in the ...
3
votes
1answer
67 views

Proving that if $n>2$ then $n!>n^{n/2}$ using induction. [duplicate]

How to prove that if $n>2$ then $n!>n^{n/2}$ using induction?
3
votes
2answers
99 views

Determining whether a number is a 'factorial number'

Let's define $\mathbb{F}=\{x \in \mathbb{Z^+}: x=n!, n=\mathbb{N}\}$ to be the set of all 'factorial numbers' (i.e. all positive integers which are the factorial of some natural number). Is there ...
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2answers
51 views

How to reduce large combinations?

The result of a hypergeometric distribution question that I posted about earlier this evening is what follows: $$\frac{{30 \choose 10}{20 \choose 5}}{{50 \choose 15}}$$ This becomes: ...
0
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2answers
24 views

Factorial Summation Problem [duplicate]

$$\sum_{j=0}^n j\cdot j!$$ I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
1
vote
1answer
50 views

Complexity of computing $N!$

Question: Complexity of computing $N!$, considering that each multiplication cost about $O(\log^2{n})$. Attempt: There's $n-1$ multiplication. Each multiplication leads to a bigger number, thus $n-1$ ...
1
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2answers
39 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
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2answers
24 views

Represent non-integer values on the factorial base

I want to compute the representation of the following values using the factorial number system: $\pi$ $e$ $\phi$ I know how to do it for integer values, but is it feasible for non-integer values? ...
5
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1answer
185 views

Finding every triplet $(n,a,b)$ such that $n!=2^a-2^b$

Question : Let $n,a,b$ be positive integers. Are there infinitely many triplets $(n,a,b)$ which satisfy the following equality?$$n!=2^a-2^b$$ If Yes, then how can we prove that? If No, then how ...
2
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1answer
84 views

Challenge: How to prove this reduction identity for factorials of even numbers?

Some time ago, as a by-product of a proof, I came across an odd (at least to me) identity for reducing the factorial of an even number into a sum: $$(2n)!=\sum_{k=0}^{\lfloor \frac{n}2 \rfloor} ...
2
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3answers
96 views

Another question about ratios of Pochhammer symbols

My question is similar to this question. Can $$\frac{(11/6)_n (7/6)_n (3/2)_n}{(3)_n}$$ be expressed 'nicely' in terms of factorials just like $(1/6)_n (1/2)_n (5/6)_n$ in the aforementioned question? ...
0
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2answers
35 views

Factorial Equivalencies Calculus

Why is: (2n+1)! = (2n)(2n+1)(2n-1)! Using this, I can deduce that: (n+1)! = (n)(n+1)(n-1)! I am working with Calculus ...
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2answers
48 views

Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in ...
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8answers
169 views

Telescoping series of form $\sum (n+1)\cdot…\cdot(n+k)$

Wolfram Alpha is able to telescope sums of the form $\sum (n+1)\cdot...\cdot(n+k)$ e.g. $(1\cdot2\cdot3) + (2\cdot3\cdot4) + (...) + n(n+1)(n+2)$ How does it do it? EDIT: We can rewrite as: $\sum ...
0
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1answer
14 views

Coefficients of a polynomial representation of factorials

I'm trying to figure out the coefficients of $${(k+d)!}/{(k-n)!}$$ when expressed as a polynomial. Any ideas?
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0answers
27 views

what is meant by : np0+np1+np2?

n+1! / n-1! = 72 (n+1)(n)(n-1!) / (n-1!) = 72 (n+1)(n) = 72 n^2 + n - 72 = 0 n = -9 (refused) , n = 8 i need an answer , this is my first math home work this year and i was able to find n as ...
2
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3answers
120 views

Proving that $(xy)!/y!^x$ is an integer

I'm learning about factorials and combinatorics in class, and this problem came up, but I don't know how to solve it. The teacher said that it would be an integer, but how can I show this? $$ ...
6
votes
4answers
174 views

Closed-forms of infinite series with factorial in the denominator

How to evaluate the closed-forms of series \begin{equation} 1)\,\, \sum_{n=0}^\infty\frac{1}{(3n)!}\qquad\left|\qquad2)\,\, \sum_{n=0}^\infty\frac{1}{(3n+1)!}\qquad\right|\qquad3)\,\, ...
4
votes
4answers
199 views

Finding $\frac{\mathrm d}{\mathrm dx} x!$

I'm trying to differentiate $x!$ but I just can't seem to do it right. I define the function as follows: $$x! = \prod_{r = 0}^{x}(x-r) \quad,\quad x \in \mathbb N$$ I've tried attempted to try it by ...
1
vote
4answers
51 views

How to find the remainder when the following series is divided by 12? [duplicate]

$1! + 2! + 3!+\cdots + 99! + 100!$ I am not getting any idea on how to solve this problem. I know that modular arithmetic should be used but not getting how to start off with the solution. Please ...
1
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0answers
27 views

Counting zeros in a factorial(terminal + zeros in between digits)

The usual counting zeros in a factorial asks to count only the terminal zeros.This question, which also asks to count the zeros that are in between digits,for example, 8! (40320, has a zero between 4 ...
1
vote
1answer
63 views

Prove $1! + 2! + . . . + n! < (n + 1)!$ using mathematical induction [duplicate]

$1! + 2! + . . . + n! < (n + 1)!$ This question has left me stumped for quite some time. I am not sure how to approach it. (I am really bad at induction).
2
votes
1answer
114 views

Use induction to prove the following: $1! + 2! + … + n! \le (n + 1)!$

Use induction to prove the following: $1! + 2! + .... + n! < (n + 1)!$ Base case: $n = 1$ $1! < 2!$ true Inductive step: Assume that $1! + 2! + .... + k! \le (k + 1)!$ is true let $n = k ...
16
votes
3answers
272 views

Approximating $100!$

I participated in an Estimathon (a speed contest of Fermi problems) not long ago. It works as follows. Contestants are given questions and they must give a closed range $[a,b]$ which should contain ...